Problem: Find an explicit formula for the geometric sequence $3\,,\,15\,,\,75\,,\,375,...$. Note: the first term should be $\textit{a(1)}$. $a(n)=$
Answer: In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{375}{75}=\dfrac{75}{15}=\dfrac{15}{3}={5}$ We see that the constant ratio between successive terms is ${5}$. In other words, we can find any term by starting with the first term and multiplying by ${5}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $g(n)$ ${3}\cdot\!{5}^{\,0}$ ${3}\cdot\!{5}^{\,1}$ ${3}\cdot\!{5}^{\,2}$ ${3}\cdot\!{5}^{\,3}$ We can see that every term is the product of the first term, ${3}$, and a power of the constant ratio, ${5}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (Note that ${3}$ is the first term and ${5}$ is the constant ratio): $a(n)={3}\cdot{5}^{{\,n-1}}$ Note that this solution strategy results in this formula; however, an equally correct solution can be written in other equivalent forms as well.